Three Experimental Modeling Systems , by Francis Muir

Three experimental dynamical models of computation, M1, M2 and M3, supplement the lattice gas as alternatives to schemes based on PDE's. These models share a common structure with the lattice gas, and may solve similar problems; they differ substantially in how they represent data, and in the sharply defined principle that controls collision rule selection. M1, like the lattice gas, is a stochastic model. The model field is a set of realizations of a random process. Field elements are binary, discretized in time, space, and velocity, and are initially selected with probability depending on the data value. Unlike the lattice gas, space and time sampling is not necessarily fine, and the crude representation of the data in any one copy is compensated by the number of copies. Evolution proceeds by translation and collision; the latter, probabilistic and irreversible, based on an entropy principle. The second model, M2, replaces M1's set of binary fields with an ensemble-average of such fields, and the field elements now range from zero to unity. At each step and at each point, these averages are disassembled into their component (M1-like) binary vectors, which are then replaced by constraint-equivalent averages, and re-assembled. The third model, M3, specializes M2 for applications where the initial conditions are close to equilibrium; the problem is now linear, and the field consists of vectors of real numbers with the collision process replaced by an idempotent matrix operation on the velocity vectors. While none of the three model systems is reversible, all have stable inverses. M2 and M3 preserve most of the attractive features of the lattice gas, and may generate very efficient and noise-free schemes, since they represent the data economically and have no random element.


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